Example 4: Power and Sample Size in Complex
Factorial ANOVA
STATISTICA Power Analysis includes several
distribution calculators that have a wide range of potential uses. The
current version of the program includes specialized dialogs for calculating
power and sample size in 1Way and 2Way factorial analysis
of variance designs. Future versions of the program may include additional
specialized analyses for more complicated ANOVA designs, but suppose you
need to calculate power for the test of a main effect in a 3way 2x2x4
factorial ANOVA now. In this exercise, we show how you can calculate the
power using the noncentral F
distribution calculator. Our approach will allow you to compute power
for any effect in a completely randomized factorial ANOVA, no matter how
complicated, in a matter of a moment or two.
Calculating the Degrees
of Freedom. Select Power Analysis from the Statistics menu to display the Power Analysis and Interval Estimation
Startup Panel. From the Startup Panel, select Probability
Distributions and Noncentral
F Distribution.
Now, click the OK button to
display the Noncentral F Probability Calculator dialog.
Note that there are three parameters in a noncentral F distribution, numerator
degrees of freedom, denominator degrees of freedom, and the noncentrality
parameter delta.
You can obtain the degrees of freedom for your design in a number of
ways. Suppose your design has pvalues
of factor A, qvalues of factor B, and rvalues of factor C. A number
of ANOVA textbooks give the degrees of freedom for this design. For example,
Kirk (1995) gives the degrees of freedom formulae shown in the table below.
Source 
Df 
A 
p  1 
B 
q  1 
C 
r  1 
AB 
(p  1) ( q  1) 
AC 
(p  1) ( r  1) 
BC 
(q  1) ( r  1) 
ABC 
(p  1) (q  1) (r  1) 
Within 
pqr (N  1) 
Suppose we want to compute the power for a test of the C main effect
in a 2x2x4 design, when the sample size per cell is N = 8. The numerator
degrees of freedom are those for the Source
in the above table, while the denominator degrees of freedom in this (fixedeffects)
ANOVA is always equal to those for Within.
Hence the degrees of freedom for the numerator are 4  1 = 3, and degrees
of freedom for the denominator are (2)(2)(4)(7) = 112.
Calculating the Noncentrality
Parameter. Simple formulae for the noncentrality parameter in F tests of significance in ANOVA are
much more difficult to find than the corresponding formulae for degrees
of freedom. Steiger and Fouladi (1997) give a simple formula that expresses
the relationship between the RMSSE, or "root
mean square standardized effect" and the noncentrality parameter
for tests of
significance in completely randomized factorial ANOVA designs. Their formula
is


(4) 
In this formula, dfeffect
is the numerator degrees of freedom parameter for the effect being tested,
and neffect
is the aggregate sample size for the means being compared in the test
of the particular effect. For example, consider a 3x3 analysis of variance.
A test of a "row main effect" involves comparing three row means
for equality. These three row means are based on sample sizes of 3N =
24 each, because they are aggregated across the three levels of the other
factor. In general, when sample size per cell is equal, one computes neffect
the integer resulting from the division of the total experiment size by
all levels of subscript contained in that term. (If all factors are involved
in the effect, neffect
=
N.)
In our current example of a 2x2x4 ANOVA, dfeffect
=
3, and neffect
= (2)(2)(8) =32.
In the present case of a three way ANOVA that has pvalues of factor
A, qvalues of factor B, and rvalues of factor C, neffect
values are as given in the following table.
Source 
Levels 
neffect 
A 
p 
Nqr 
B 
q 
Npr 
C 
r 
Npq 
AB 

Nr 
AC 

Nq 
BC 

Np 
ABC 

N 
By simply squaring both sides and rearranging Equation 4, we have
effect
=
neffectdfeffectRMSSE2effect 

(5) 
As we saw in a previous
example, an RMSSE of approximately .30 would correspond to what Cohen
(1983) would call "medium effects" in this design. Using that
value for RMSSE, we obtain a value for the noncentrality parameter of
(3)(32)(.03)2
=
8.64.
Completing the Power Calculation.
Once values for the three parameters are available, it becomes a relatively
trivial matter to compute power. The power calculation will proceed in
three steps:
Set
the correct degrees of freedom.
Calculate
the critical value of F, using a noncentrality parameter of 0. Call this
value Fcrit.
Set
the noncentrality parameter to the appropriate value corresponding to
the effect size in the population. Compute 1  p for Fcrit.
This is the power for the ANOVA Ftest
Return your attention to the Noncentral F Probability Calculator  Quick
tab. Change the Numerator df
to 3, and the Denom.
df to 112. To calculate
the critical value of F with
an = .05, select
the (1  Cumulative p) check
box, and check that 0.05 is the
value for 1  Cum. p.
In the Compute
group, select F as the quantity
to compute. You are now set up to compute the critical value of the F3,112 distribution.
Click the Compute button, and
the critical value appears in the Observed
F box.
You should observe a critical value of 2.685643. To calculate power for the
Ftest, we simply compute the
proportion of the probability in a noncentral F3,112,
8.64 distribution. Enter 8.64
as the value for Delta, and click
the 1  p option button under
Compute as the value to compute.
Then click the Compute button.
The dialog should then appear as below.
You have thus determined the power to be approximately .67.
In a similar vein, suppose you have a 2x3x2x4 factorial ANOVA, and wish
to determine the power for the ABD interaction when the RMSSE is .3, and
N =3. Denoting the levels of factors A,B,C,D as p,q,r,s, we find that
the numerator degrees of freedom parameter for the ABD interaction effect
is
dfABD = (p
 1)(q  1)(s  1) = (2  1)(3  1)(4  1) = 6
The denominator degrees of freedom are
dfwithin =
pqrs(N  1) = (2)(3)(2)(4)(3  1) = 96
The means used in computing the interaction effect are aggregated across
the two levels of factor D, so we have
neffect =
2N = 6
Consequently, the noncentrality parameter is
ABD
= (6)(6)(.3)2 = 3.24
Switching to the probability calculator,
we first obtain the critical value of the Fstatistic,
by entering the degrees of freedom (6
and 96), setting 1
 Cum. p. to .05, selecting
F as the quantity to Compute,
and clicking the Compute button.
The result should look like this:
Next, compute the proportion of a noncentral F
with the same degrees of freedom, but a noncentrality parameter of 3.24,
that falls above the critical value just established. Simply enter 3.24 as Delta,
click on the 1  p option button
to change the computed quantity, and click the Compute
button. The result should look like this:
Unfortunately, the power is a meager .208, and clearly this design will
not allow you to test the ABD interaction effectively with the planned
sample size.
See also, Power
Analysis  Index.